Pathways to relativistic curved momentum spaces: de Sitter case study

TL;DR

This paper explores the structure of curved momentum spaces with de Sitter geometry, proposing a new way to associate affine connections with momentum composition laws, leading to novel DSR-relativistic models with commutative momentum laws.

Contribution

It introduces a new prescription for associating affine connections to momentum composition, expanding the understanding of de Sitter momentum spaces and their relativistic properties.

Findings

Two prescriptions for affine connections yield the same $oldsymbol{ ext{kappa}}$-momentum space.

A new approach produces a proper de Sitter momentum space with a commutative composition law.

The study offers a test case for momentum spaces with commutative yet deformed composition laws.

Abstract

Several arguments suggest that the Planck scale could be the characteristic scale of curvature of momentum space. As other recent studies we assume that the metric of momentum space determines the condition of on-shellness while the momentum-space affine connection governs the form of the law of composition of momenta. We show that the possible choices of laws of composition of momenta are more numerous than the possible choices of affine connection on a momentum space. This motivates us to propose a new prescription for associating an affine connection to momentum composition, which we compare to the one most used in the recent literature. We find that the two prescriptions lead to the same picture of the so-called $\kappa$-momentum space, with de Sitter metric and $\kappa$-Poincar\'e connection. We also examine in greater detail than ever before the DSR-relativistic properties of $\kappa$-momentum space, particularly in relation to its noncommutative law of composition of momenta. We then show that in the case of "proper de Sitter momentum space", with the de Sitter metric and its Levi-Civita connection, the two prescriptions are inequivalent. Our novel prescription leads to a picture of proper de Sitter momentum space which is DSR-relativistic and is characterized by a commutative law of composition of momenta, a possibility for which no explicit curved-momentum-space picture had been previously found. We argue that our construction provides a natural test case for the study of momentum spaces with commutative, and yet deformed, laws of composition of momenta. Moreover, it can serve as laboratory for the exploration of the properties of DSR-relativistic theories which are not connected to group-manifold momentum spaces and Hopf algebras.